Second-order logic and foundations of mathematics. (English) Zbl 1002.03013

Summary: We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically different: the latter is a major fragment of the former.


03B30 Foundations of classical theories (including reverse mathematics)
03B15 Higher-order logic; type theory (MSC2010)
03E99 Set theory
03A05 Philosophical and critical aspects of logic and foundations
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